To put you in prison if they find by some mean you did.

The mean could be a 5$ wrench, hacking into your devices or plain old surveillance

If you lie, they can either put you in prison or deport you if you are not a citizen.

If I lie, and they can prove that I lied (e.g. I organized illegal activities, which is itself illegal) they can do that anyway.

There's absolutely nothing you gain by making it easy for them by telling them that you did.

Criminals tell on themselves every day. Whether that's the truth or not isn't actually to cops.